Having played and coached youth sports in the past, I can say with some credibility that Pop Warner football, Little League baseball, etc. are doing a lot more harm than good. The exposure to physical injuries is only a very small part of the problem.
“Back in the day” we would play sandlot *everything* (well, almost — that would be tough on the skates ;). Find an open field, self-organize w/o adult interference, everyone played...
Now with the lack of available space, more things for kids to do & the nanny state ready to pounce if Johnny gets so much as a scratch, sadly those times are probably gone.